Symbols and Complex Numbers

Symbolic variables#

Symbolic variables, called symbols, must be defined and assigned to Python variables before they can be used. This is typically done with the Symbol function:

x = Symbol('x' )

To assign multiple symbols in a single function call, we use the symbols function:

x, y, z = symbols('x y z')

creates three symbols representing variables named x, y, and z. In this particular instance, these symbols are all assigned to Python variables of the same name. However, the user is free to assign them to different Python variables, while representing the same symbol, such as a, b, c = symbols('x y z').

It is good practice to have the same names for variables and their associated symbols.

We can explicitly set the properties positive, real, imaginary and complex for the symbol. This helps us simplify the expressions.

x = Symbol('x', real = True, positive = True)

Use the help(Symbol) command to find more properties.

Complex numbers#

SymPy never performs simplifications on an expression unless their properties allow it.

For instance, the simplification $\sqrt{x^2}=x$ holds if $x$ is nonnegative ($x \geq 0$), but it does not hold for a general complex $x$.

By default, SymPy performs all calculations assuming that symbols are complex. This makes it easier to solve mathematical problems regardless of whether a symbol is complex or not. Let’s look at an example of this below:

Intuitively the answer to this should have been $x$, but by assuming the most general case, that $x$ is complex by default, SymPy avoids performing mathematically invalid operations.

The imaginary unit is denoted by I in SymPy.

Conversion to Python complex#

The type of complex number generated from I is sympy.core.add.Add. We can use the built-in complex function to convert a SymPy complex to the regular Python complex type.

We will learn about numerical evaluation in the next lesson.